Is it permissible for my wife to remove the niqab, which she vowed to wear if she became pregnant, after she became unable to breathe because of it, or would that be considered a breach of her promise and vow, for which she would be sinful?

Covering the face is obligatory according to the more preponderant opinion, and its obligation is further emphasized by a vow. However, a vow to perform something already obligatory is not valid according to the majority of scholars, because it is a foregone conclusion. Therefore, nothing is binding as a result of this vow, but it is obligatory to adhere to the niqab. It is not permissible to uncover the face in front of non-mahram (definition) men, unless covering it would genuinely cause harm and there is no alternative, in which case it is permissible to uncover it to the extent of the necessity.